Slice ginger, lime leaves, green onions, lemongrass, garlic, and jalapeño. Place in a mason jar and cover with sesame oil. Shake well and refrigerate overnight, or up to several days for best flavor.
Pour oil and 1/2 ginger, lime leaves, green onions, garlic, and jalapeño into a blender. Add coconut milk and juice of 2 limes. Blend until smooth.